How narrow is the sQGP transition? A simple non-perturbative - PowerPoint PPT Presentation

At T >> T c a gas of Debye screened gluons: l D ∼ gT >> 1 1 T Any gluon species with charge ± 1: contributes exp ( i 2 π/ 2 ) = − 1. In the slab are on average ¯ l = n ( T ) l D . Area gluons of hat species. Poisson distribution for average due to a charged species: l l l ! ( − 1 ) l exp ( − ¯ ¯ < V k > one cs = � l ) = exp ( − 2 ¯ l ) l All 2 k ( N − k ) charged gluon species (supposed independent): < V k > = exp ( − 4 k ( N − k ) l D n ( T ) . Area ) Casimir scaling: ρ k ( T ) ∼ k ( N − k ) l D n ( T ) How narrow is the sQGP transition?

At T >> T c a gas of Debye screened gluons: l D ∼ gT >> 1 1 T Any gluon species with charge ± 1: contributes exp ( i 2 π/ 2 ) = − 1. In the slab are on average ¯ l = n ( T ) l D . Area gluons of hat species. Poisson distribution for average due to a charged species: l l l ! ( − 1 ) l exp ( − ¯ ¯ < V k > one cs = � l ) = exp ( − 2 ¯ l ) l All 2 k ( N − k ) charged gluon species (supposed independent): < V k > = exp ( − 4 k ( N − k ) l D n ( T ) . Area ) Casimir scaling: ρ k ( T ) ∼ k ( N − k ) l D n ( T ) How narrow is the sQGP transition?

At T >> T c a gas of Debye screened gluons: l D ∼ gT >> 1 1 T Any gluon species with charge ± 1: contributes exp ( i 2 π/ 2 ) = − 1. In the slab are on average ¯ l = n ( T ) l D . Area gluons of hat species. Poisson distribution for average due to a charged species: l l l ! ( − 1 ) l exp ( − ¯ ¯ < V k > one cs = � l ) = exp ( − 2 ¯ l ) l All 2 k ( N − k ) charged gluon species (supposed independent): < V k > = exp ( − 4 k ( N − k ) l D n ( T ) . Area ) Casimir scaling: ρ k ( T ) ∼ k ( N − k ) l D n ( T ) How narrow is the sQGP transition?

At T >> T c a gas of Debye screened gluons: l D ∼ gT >> 1 1 T Any gluon species with charge ± 1: contributes exp ( i 2 π/ 2 ) = − 1. In the slab are on average ¯ l = n ( T ) l D . Area gluons of hat species. Poisson distribution for average due to a charged species: l l l ! ( − 1 ) l exp ( − ¯ ¯ < V k > one cs = � l ) = exp ( − 2 ¯ l ) l All 2 k ( N − k ) charged gluon species (supposed independent): < V k > = exp ( − 4 k ( N − k ) l D n ( T ) . Area ) Casimir scaling: ρ k ( T ) ∼ k ( N − k ) l D n ( T ) How narrow is the sQGP transition?

At T >> T c a gas of Debye screened gluons: l D ∼ gT >> 1 1 T Any gluon species with charge ± 1: contributes exp ( i 2 π/ 2 ) = − 1. In the slab are on average ¯ l = n ( T ) l D . Area gluons of hat species. Poisson distribution for average due to a charged species: l l l ! ( − 1 ) l exp ( − ¯ ¯ < V k > one cs = � l ) = exp ( − 2 ¯ l ) l All 2 k ( N − k ) charged gluon species (supposed independent): < V k > = exp ( − 4 k ( N − k ) l D n ( T ) . Area ) Casimir scaling: ρ k ( T ) ∼ k ( N − k ) l D n ( T ) How narrow is the sQGP transition?

Reduced electric flux tension in deconfined phase 2.2 2 1.8 1.6 σ k /T 2 / (k (N-k)) SU(3) 1.4 SU(4), k=1 SU(4), k=2 1.2 SU(6), k=1 SU(6), k=2 1 SU(6), k=3 SU(8), k=1 0.8 SU(8), k=2 SU(8), k=3 0.6 SU(8), k=4 GKA T/ Λ MSbar =1.35 0.4 1 1.5 2 2.5 3 3.5 4 4.5 T/T c N c ≤ 8, de Forcrand et al.,hep-lat/0510081, Bursa/Teper, hep-lat/0505025 GKA: field theory calculation to two loop order hep-ph/0102022, cubic order in hep-ph0412322. BGKAP: PRL66, 998, 1991. How narrow is the sQGP transition?

Electric flux tension in the deconfined phase e -tension for SU ( N c ) , N c ≤ 8, PdF et al.,hep-lat/051008 How narrow is the sQGP transition?

Electric flux tension Casimir scaling good for ANY T above 1.15 T c in deconfined phase Two loop reduced tension does not match the lattice calculation Warranted: 3 or more loop calculation (Yannis Burnier, ‘CPKA, York Schroeder, Aleksi Vuorinen). This talk: perhaps a more insightful way -beyond perturbation theory- to understand the Casimir scaling down to ≥ T c How narrow is the sQGP transition?

Casimir scaling of spatial Wilson loops GKA (hep -ph0102022): at high T a dilute gas of adjoint monopoles causes Casimir scaling for Wilson loops. Lucini Teper (2001....) and hep-lat/051008 : How narrow is the sQGP transition?

Casimir scaling of spatial Wilson loops GKA (hep -ph0102022): at high T a dilute gas of adjoint monopoles causes Casimir scaling for Wilson loops. Lucini Teper (2001....) and hep-lat/051008 : How narrow is the sQGP transition?

Casimir scaling of spatial Wilson loops GKA (hep -ph0102022): at high T a dilute gas of adjoint monopoles causes Casimir scaling for Wilson loops. Lucini Teper (2001....) and hep-lat/051008 : How narrow is the sQGP transition?

m -flux tension at asymptotic T. Lattice results Binding energy of k-strings 9 8 7 6 k=2 (k- σ k / σ 1 )N k=3 5 Casimir scaling Sine formula 4 3 2 1 0 0 0.05 0.1 0.15 0.2 0.25 1/N flux-tension for SU ( N c ) , N c ≤ 8, Meyer, hep-lat/0412021) How narrow is the sQGP transition?

m -flux tension at asymptotic T The k=N/2 string tension 1 0.8 2 σ k=N/2 / (N σ 1 ) 0.6 0.4 0.2 Casimir scaling Sine formula 0 0 0.05 0.1 0.15 0.2 0.25 1/N m-tension for SU ( N c ) , N c ≤ 8, Meyer, hep-lat/0412021 How narrow is the sQGP transition?

3d results propagate in ALL of the deconfined phase through the running coupling _ T c / Λ MS = 1.10...1.35 1.2 2-loop 1.0 1/2 T/ σ s 1-loop 0.8 1 4d lattice, N τ = 8 0.6 1.0 2.0 3.0 4.0 5.0 T / T c m -tension, N c = 3, hep-lat/0503003 σ ( T ) = c 3 d g 4 M ( T ) = c 3 d g 4 E ( T )( 1 + small ) = c 3 d g 4 ( T )( 1 + .. + 3 loop ) T 2 How narrow is the sQGP transition?

How do magnetic and electric flux compare? � � 2.0 � � � � � � � � 1.8 �� �� �� �� �� �� �� �� �� �� �� �� �� �� 1.6 � � � � � � 1−4 � � � � �� �� 1.2 �� �� �� �� �� �� �� �� 1.0 �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� 0.8 � � � � � � 0.6 �� �� � � �� �� � � �� �� �� �� �� �� �� �� �� �� � � � � � � �� �� � � 0.4 l l l �� �� � � � � �� �� T/T c 1 2 3 4 SU(3) colour electric flux versus SU(3) colour magnetic flux Note: equality inside the peak of the interaction measure T / T c ∼ 1 . 10. So peak might be due to a correlation of electric and magnetic quasi -particles How narrow is the sQGP transition?

Correlations between loops Measure correlation on the lattice between nearby, almost contingent ’t Hooft and Wilson loop as function of temperature. For very high T: magnetic and electric populations are uncorrelaled, so expect no correlation between loops. For T in critical region around the peak of the conformal energy the correlation may become quite strong. The correlation is a key quantity for understanding the behaviour of the plasma components. Unfortunately it is subleading in in large N limit, so simplest AdS/CFT is not enought to access it. How narrow is the sQGP transition?

The ratio δ as function of T. SU(3) case 2.1 _ \delta X _ 1.8 1.5 _ x 1.2 _ 0.9 _ x _ 0.6 x _ 0.3 x x x x x x x x | | | T/T_c 0 1 2 3 00 \delta=\sigma_s/(m_0^++)^2, colours as in previous figure. The ratio σ 1 / m 2 ++ , SU(3), Datta, Gupta,hep -lat/0208001 How narrow is the sQGP transition?

At T ∼ 1 . 2 T c the ratio has risen with a factor 10 From large T to T c the ratio increases with a factor 40! ! SU(3) weakly first order, may explain the large ratio. m −− is probably the inverse radius of the adjoint magnetic quasi particle, determines a much smaller ratio which would be the diluteness l 3 −− n M , but is not yet available for all T. How narrow is the sQGP transition?

Perturbation theory and the flux loops Once the non -perturbative 3d part of the magnetic loops is detemined on lattice, perturbation theory works, and they have Casimir scaling. Although the magnetic free energy scales as a gas of adjoint quasi-particles, no classical adjoint monopoles are known in QCD. The electric loops have Casimir scaling according to one two and two loop order. To three loop order the preliminary results suggest the same. "Precocious" QGP behaviour (see below) may be an alternative explanation How narrow is the sQGP transition?

Perturbation theory and the flux loops Once the non -perturbative 3d part of the magnetic loops is detemined on lattice, perturbation theory works, and they have Casimir scaling. Although the magnetic free energy scales as a gas of adjoint quasi-particles, no classical adjoint monopoles are known in QCD. The electric loops have Casimir scaling according to one two and two loop order. To three loop order the preliminary results suggest the same. "Precocious" QGP behaviour (see below) may be an alternative explanation How narrow is the sQGP transition?

Perturbation theory and the flux loops Once the non -perturbative 3d part of the magnetic loops is detemined on lattice, perturbation theory works, and they have Casimir scaling. Although the magnetic free energy scales as a gas of adjoint quasi-particles, no classical adjoint monopoles are known in QCD. The electric loops have Casimir scaling according to one two and two loop order. To three loop order the preliminary results suggest the same. "Precocious" QGP behaviour (see below) may be an alternative explanation How narrow is the sQGP transition?

Perturbation theory and the flux loops Once the non -perturbative 3d part of the magnetic loops is detemined on lattice, perturbation theory works, and they have Casimir scaling. Although the magnetic free energy scales as a gas of adjoint quasi-particles, no classical adjoint monopoles are known in QCD. The electric loops have Casimir scaling according to one two and two loop order. To three loop order the preliminary results suggest the same. "Precocious" QGP behaviour (see below) may be an alternative explanation How narrow is the sQGP transition?

Perturbation theory and the flux loops Once the non -perturbative 3d part of the magnetic loops is detemined on lattice, perturbation theory works, and they have Casimir scaling. Although the magnetic free energy scales as a gas of adjoint quasi-particles, no classical adjoint monopoles are known in QCD. The electric loops have Casimir scaling according to one two and two loop order. To three loop order the preliminary results suggest the same. "Precocious" QGP behaviour (see below) may be an alternative explanation How narrow is the sQGP transition?

Field theory calculation of loop average � V k ( L ) � = Tr phys V k ( L ) exp ( − H / T ) / Tr phys exp ( − H / T ) By translation into path integral language: Domainwall at z=0 between domains where Polyakov loop takes different Z(N) values has energy ρ k ( T ) . 2 z 1 > 3 z=0 x 0 qY k t P=exp(ik2 π/Ν) k N−1 P=1 Mimima of effective potential Periodic time direction and z−direction orthogonal numbered by k to (x,y) plane. Loop L x L at z=0. x y Polyakov loop profile along z -direction, and Z(N) vacua. Effective action: U = K ( q ) q ′ 2 + V ( q ) in loop expansion. tunnneling along qY k between the minima gives ρ k energy of wall/per unit length= ρ k ( T ) . How narrow is the sQGP transition?

Field theory calculation of loop average � V k ( L ) � = Tr phys V k ( L ) exp ( − H / T ) / Tr phys exp ( − H / T ) By translation into path integral language: Domainwall at z=0 between domains where Polyakov loop takes different Z(N) values has energy ρ k ( T ) . 2 z 1 > 3 z=0 x 0 qY k t P=exp(ik2 π/Ν) k N−1 P=1 Mimima of effective potential Periodic time direction and z−direction orthogonal numbered by k to (x,y) plane. Loop L x L at z=0. x y Polyakov loop profile along z -direction, and Z(N) vacua. Effective action: U = K ( q ) q ′ 2 + V ( q ) in loop expansion. tunnneling along qY k between the minima gives ρ k energy of wall/per unit length= ρ k ( T ) . How narrow is the sQGP transition?

Field theory calculation of loop average � V k ( L ) � = Tr phys V k ( L ) exp ( − H / T ) / Tr phys exp ( − H / T ) By translation into path integral language: Domainwall at z=0 between domains where Polyakov loop takes different Z(N) values has energy ρ k ( T ) . 2 z 1 > 3 z=0 x 0 qY k t P=exp(ik2 π/Ν) k N−1 P=1 Mimima of effective potential Periodic time direction and z−direction orthogonal numbered by k to (x,y) plane. Loop L x L at z=0. x y Polyakov loop profile along z -direction, and Z(N) vacua. Effective action: U = K ( q ) q ′ 2 + V ( q ) in loop expansion. tunnneling along qY k between the minima gives ρ k energy of wall/per unit length= ρ k ( T ) . How narrow is the sQGP transition?

Field theory calculation of loop average � V k ( L ) � = Tr phys V k ( L ) exp ( − H / T ) / Tr phys exp ( − H / T ) By translation into path integral language: Domainwall at z=0 between domains where Polyakov loop takes different Z(N) values has energy ρ k ( T ) . 2 z 1 > 3 z=0 x 0 qY k t P=exp(ik2 π/Ν) k N−1 P=1 Mimima of effective potential Periodic time direction and z−direction orthogonal numbered by k to (x,y) plane. Loop L x L at z=0. x y Polyakov loop profile along z -direction, and Z(N) vacua. Effective action: U = K ( q ) q ′ 2 + V ( q ) in loop expansion. tunnneling along qY k between the minima gives ρ k energy of wall/per unit length= ρ k ( T ) . How narrow is the sQGP transition?

A non -perturbative approach I_3 domain at infinite T ω * domain at T_c 0 Y ω !/3 TrP=1 Domain of the SU(3) effective potential in Cartan space Infinite T see perturbative potential T ≥ T c see histogram Thermodynamic functions live on the C invariant minima (red lines, Z(3) related copies) we want a model for the potential V in between these temperatures. we can compute the tunneling between Z(3) related vacua How narrow is the sQGP transition? (e flux tension) or the tunneling from V at T to TrP 0,

A non -perturbative approach I_3 domain at infinite T ω * domain at T_c 0 Y ω !/3 TrP=1 Domain of the SU(3) effective potential in Cartan space Infinite T see perturbative potential T ≥ T c see histogram Thermodynamic functions live on the C invariant minima (red lines, Z(3) related copies) we want a model for the potential V in between these temperatures. we can compute the tunneling between Z(3) related vacua How narrow is the sQGP transition? (e flux tension) or the tunneling from V at T to TrP 0,

A non -perturbative approach I_3 domain at infinite T ω * domain at T_c 0 Y ω !/3 TrP=1 Domain of the SU(3) effective potential in Cartan space Infinite T see perturbative potential T ≥ T c see histogram Thermodynamic functions live on the C invariant minima (red lines, Z(3) related copies) we want a model for the potential V in between these temperatures. we can compute the tunneling between Z(3) related vacua How narrow is the sQGP transition? (e flux tension) or the tunneling from V at T to TrP 0,

A non -perturbative approach I_3 domain at infinite T ω * domain at T_c 0 Y ω !/3 TrP=1 Domain of the SU(3) effective potential in Cartan space Infinite T see perturbative potential T ≥ T c see histogram Thermodynamic functions live on the C invariant minima (red lines, Z(3) related copies) we want a model for the potential V in between these temperatures. we can compute the tunneling between Z(3) related vacua How narrow is the sQGP transition? (e flux tension) or the tunneling from V at T to TrP 0,

0 . 8 0 . 6 V eff 0 . 4 0 . 2 0 1.0 0.5 Im[p] 0.0 -0.5 -1.0 -0.5 0.0 0.5 Re[p] 1.0 SU(3), lowest order perturbative effective potential. How narrow is the sQGP transition? 3) minima: gulleys of least action along the boundary of the domain (hypercha

0 . 1 5 Im Ω 0 -0.15 0 - 0 . 1 5 0.15 Re Ω Histogram of the Polyakov loop P in SU(3). It equals exp − ( Vol ) V ( P ) . The Z(3) minima have moved in towards the symmetric point. the Z(3) symmetric point a new minimum is developing. T c when all degenera How narrow is the sQGP transition?

A non perturbative approach to the effective potential Intiated by Meisinger, Miller and Ogilvie hep -phys/0109009 and 0108026 Motivated by a remark of RDP at Kyoto 2006 Idea is to make an Ansatsz for V that consists of Z(N) symmetric "trial functions": B p ( P ) = � l w l Tr adj P ( A 0 ) l . Simplest are those with w l = 1 / l p , p = 4 , 2. Are corresponding to the fluctuation determinant, resp tadpole of the gluon. Correspond to simple Bernoulli polynomials: � � P diag exp ( i 2 π q 1 ) , .........., exp ( i 2 π q N ) = | q i − q j | p ( 1 − | q i − q j | ) p , B 2 p ( P ) � ∼ i , j Perturbative answer is B 4 , minima at q i − q j == 0 mod 1. To destabilize those minima: need linear term in q i − q j , and the unique candidate is B 2 with a negative coefficient. How narrow is the sQGP transition?

A non perturbative approach to the effective potential Intiated by Meisinger, Miller and Ogilvie hep -phys/0109009 and 0108026 Motivated by a remark of RDP at Kyoto 2006 Idea is to make an Ansatsz for V that consists of Z(N) symmetric "trial functions": B p ( P ) = � l w l Tr adj P ( A 0 ) l . Simplest are those with w l = 1 / l p , p = 4 , 2. Are corresponding to the fluctuation determinant, resp tadpole of the gluon. Correspond to simple Bernoulli polynomials: � � P diag exp ( i 2 π q 1 ) , .........., exp ( i 2 π q N ) = | q i − q j | p ( 1 − | q i − q j | ) p , B 2 p ( P ) � ∼ i , j Perturbative answer is B 4 , minima at q i − q j == 0 mod 1. To destabilize those minima: need linear term in q i − q j , and the unique candidate is B 2 with a negative coefficient. How narrow is the sQGP transition?

A non perturbative approach to the effective potential Intiated by Meisinger, Miller and Ogilvie hep -phys/0109009 and 0108026 Motivated by a remark of RDP at Kyoto 2006 Idea is to make an Ansatsz for V that consists of Z(N) symmetric "trial functions": B p ( P ) = � l w l Tr adj P ( A 0 ) l . Simplest are those with w l = 1 / l p , p = 4 , 2. Are corresponding to the fluctuation determinant, resp tadpole of the gluon. Correspond to simple Bernoulli polynomials: � � P diag exp ( i 2 π q 1 ) , .........., exp ( i 2 π q N ) = | q i − q j | p ( 1 − | q i − q j | ) p , B 2 p ( P ) � ∼ i , j Perturbative answer is B 4 , minima at q i − q j == 0 mod 1. To destabilize those minima: need linear term in q i − q j , and the unique candidate is B 2 with a negative coefficient. How narrow is the sQGP transition?

A non perturbative approach to the effective potential Intiated by Meisinger, Miller and Ogilvie hep -phys/0109009 and 0108026 Motivated by a remark of RDP at Kyoto 2006 Idea is to make an Ansatsz for V that consists of Z(N) symmetric "trial functions": B p ( P ) = � l w l Tr adj P ( A 0 ) l . Simplest are those with w l = 1 / l p , p = 4 , 2. Are corresponding to the fluctuation determinant, resp tadpole of the gluon. Correspond to simple Bernoulli polynomials: � � P diag exp ( i 2 π q 1 ) , .........., exp ( i 2 π q N ) = | q i − q j | p ( 1 − | q i − q j | ) p , B 2 p ( P ) � ∼ i , j Perturbative answer is B 4 , minima at q i − q j == 0 mod 1. To destabilize those minima: need linear term in q i − q j , and the unique candidate is B 2 with a negative coefficient. How narrow is the sQGP transition?

A non perturbative approach to the effective potential Intiated by Meisinger, Miller and Ogilvie hep -phys/0109009 and 0108026 Motivated by a remark of RDP at Kyoto 2006 Idea is to make an Ansatsz for V that consists of Z(N) symmetric "trial functions": B p ( P ) = � l w l Tr adj P ( A 0 ) l . Simplest are those with w l = 1 / l p , p = 4 , 2. Are corresponding to the fluctuation determinant, resp tadpole of the gluon. Correspond to simple Bernoulli polynomials: � � P diag exp ( i 2 π q 1 ) , .........., exp ( i 2 π q N ) = | q i − q j | p ( 1 − | q i − q j | ) p , B 2 p ( P ) � ∼ i , j Perturbative answer is B 4 , minima at q i − q j == 0 mod 1. To destabilize those minima: need linear term in q i − q j , and the unique candidate is B 2 with a negative coefficient. How narrow is the sQGP transition?

A non perturbative approach to the effective potential Intiated by Meisinger, Miller and Ogilvie hep -phys/0109009 and 0108026 Motivated by a remark of RDP at Kyoto 2006 Idea is to make an Ansatsz for V that consists of Z(N) symmetric "trial functions": B p ( P ) = � l w l Tr adj P ( A 0 ) l . Simplest are those with w l = 1 / l p , p = 4 , 2. Are corresponding to the fluctuation determinant, resp tadpole of the gluon. Correspond to simple Bernoulli polynomials: � � P diag exp ( i 2 π q 1 ) , .........., exp ( i 2 π q N ) = | q i − q j | p ( 1 − | q i − q j | ) p , B 2 p ( P ) � ∼ i , j Perturbative answer is B 4 , minima at q i − q j == 0 mod 1. To destabilize those minima: need linear term in q i − q j , and the unique candidate is B 2 with a negative coefficient. How narrow is the sQGP transition?

Repulsive and attractive eigenvalues of the Wilson line SU(2): P = diagonal ( exp ( i φ/ 2 , exp ( − i φ/ 2 )) At high T in perturbation theory: phases cluster at centergroup values φ = 2 π q = 0 , π T 4 V pert = T 4 ( π 2 / 15 ) + 2 π 2 3 q 2 ( 1 − | q | ) 2 ) minima in q=0,1 Adding a term V nonpert = − M 2 T 2 | q | ( 1 − | q | ) induces tendency to repulsion: minima are φ = ± π So our mean field like Ansatz is: T 4 V = T 4 ( V pert + V nonpert ) 3 π 2 q 2 ( 1 − | q | ) 2 − ( M � π 2 � 15 + 2 T 4 T ) 2 ( | q | ( 1 − | q | ) + d ) = (1) At high T: the perturbative determinant term dominates: e.v.’s cluster in Z(N) As T ∼ M : the "non -perturbative" Ansatz starts to kick in: the linear term destabilizes the perturbative vacuum, e.v’s repel, equal spacing,Tr P=0, and d fixes pressure=0 at T c . How narrow is the sQGP transition?

Repulsive and attractive eigenvalues of the Wilson line SU(2): P = diagonal ( exp ( i φ/ 2 , exp ( − i φ/ 2 )) At high T in perturbation theory: phases cluster at centergroup values φ = 2 π q = 0 , π T 4 V pert = T 4 ( π 2 / 15 ) + 2 π 2 3 q 2 ( 1 − | q | ) 2 ) minima in q=0,1 Adding a term V nonpert = − M 2 T 2 | q | ( 1 − | q | ) induces tendency to repulsion: minima are φ = ± π So our mean field like Ansatz is: T 4 V = T 4 ( V pert + V nonpert ) 3 π 2 q 2 ( 1 − | q | ) 2 − ( M � π 2 � 15 + 2 T 4 T ) 2 ( | q | ( 1 − | q | ) + d ) = (1) At high T: the perturbative determinant term dominates: e.v.’s cluster in Z(N) As T ∼ M : the "non -perturbative" Ansatz starts to kick in: the linear term destabilizes the perturbative vacuum, e.v’s repel, equal spacing,Tr P=0, and d fixes pressure=0 at T c . How narrow is the sQGP transition?

Repulsive and attractive eigenvalues of the Wilson line SU(2): P = diagonal ( exp ( i φ/ 2 , exp ( − i φ/ 2 )) At high T in perturbation theory: phases cluster at centergroup values φ = 2 π q = 0 , π T 4 V pert = T 4 ( π 2 / 15 ) + 2 π 2 3 q 2 ( 1 − | q | ) 2 ) minima in q=0,1 Adding a term V nonpert = − M 2 T 2 | q | ( 1 − | q | ) induces tendency to repulsion: minima are φ = ± π So our mean field like Ansatz is: T 4 V = T 4 ( V pert + V nonpert ) 3 π 2 q 2 ( 1 − | q | ) 2 − ( M � π 2 � 15 + 2 T 4 T ) 2 ( | q | ( 1 − | q | ) + d ) = (1) At high T: the perturbative determinant term dominates: e.v.’s cluster in Z(N) As T ∼ M : the "non -perturbative" Ansatz starts to kick in: the linear term destabilizes the perturbative vacuum, e.v’s repel, equal spacing,Tr P=0, and d fixes pressure=0 at T c . How narrow is the sQGP transition?

Repulsive and attractive eigenvalues of the Wilson line SU(2): P = diagonal ( exp ( i φ/ 2 , exp ( − i φ/ 2 )) At high T in perturbation theory: phases cluster at centergroup values φ = 2 π q = 0 , π T 4 V pert = T 4 ( π 2 / 15 ) + 2 π 2 3 q 2 ( 1 − | q | ) 2 ) minima in q=0,1 Adding a term V nonpert = − M 2 T 2 | q | ( 1 − | q | ) induces tendency to repulsion: minima are φ = ± π So our mean field like Ansatz is: T 4 V = T 4 ( V pert + V nonpert ) 3 π 2 q 2 ( 1 − | q | ) 2 − ( M � π 2 � 15 + 2 T 4 T ) 2 ( | q | ( 1 − | q | ) + d ) = (1) At high T: the perturbative determinant term dominates: e.v.’s cluster in Z(N) As T ∼ M : the "non -perturbative" Ansatz starts to kick in: the linear term destabilizes the perturbative vacuum, e.v’s repel, equal spacing,Tr P=0, and d fixes pressure=0 at T c . How narrow is the sQGP transition?

Repulsive and attractive eigenvalues of the Wilson line SU(2): P = diagonal ( exp ( i φ/ 2 , exp ( − i φ/ 2 )) At high T in perturbation theory: phases cluster at centergroup values φ = 2 π q = 0 , π T 4 V pert = T 4 ( π 2 / 15 ) + 2 π 2 3 q 2 ( 1 − | q | ) 2 ) minima in q=0,1 Adding a term V nonpert = − M 2 T 2 | q | ( 1 − | q | ) induces tendency to repulsion: minima are φ = ± π So our mean field like Ansatz is: T 4 V = T 4 ( V pert + V nonpert ) 3 π 2 q 2 ( 1 − | q | ) 2 − ( M � π 2 � 15 + 2 T 4 T ) 2 ( | q | ( 1 − | q | ) + d ) = (1) At high T: the perturbative determinant term dominates: e.v.’s cluster in Z(N) As T ∼ M : the "non -perturbative" Ansatz starts to kick in: the linear term destabilizes the perturbative vacuum, e.v’s repel, equal spacing,Tr P=0, and d fixes pressure=0 at T c . How narrow is the sQGP transition?

N=2 N=3 N=4 REPULSION or SYMMETRY IS RESTORED ATTRACTION or SYMMETRY IS BROKEN As T goes down the eigenvalues start to decluster and move out to the equal spacing positions. In all but SU(2) the transition is first order, so the eigenvalues stop short of the equal spacing positions. How narrow is the sQGP transition?

N=2 N=3 N=4 REPULSION or SYMMETRY IS RESTORED ATTRACTION or SYMMETRY IS BROKEN As T goes down the eigenvalues start to decluster and move out to the equal spacing positions. In all but SU(2) the transition is first order, so the eigenvalues stop short of the equal spacing positions. How narrow is the sQGP transition?

Simple relations To determine pressure from V ( q ) : find the extrema q 0 of V ( q ) , V ′ ( q 0 ) = 0: p = − V ( q 0 ) . (2) Now the relation of ∆ to V is immediate: ∆ T ∂ � � p / T 4 = T 4 ∂ T − ∂ V ( q 0 ) = ∂ T − T ∂ q 0 ∂ T V ′ ( q 0 ) + 2 M 2 / T 2 V nonpert ( q 0 ) = (3) So ∆ relates only (not unexpected) to the non -perturbative potential.: ∆ T 4 = 2 M 2 / T 2 V nonpert ( q 0 ) . (4) How narrow is the sQGP transition?

3 orig. model A orig. model A 2.5 2 1.5 1 0.5 0 p/T 4 e/3T 4 ∆ /T 4 Latt. p/T 4 Latt. e/3T 4 Latt. ∆ /T 4 -0.5 1 1.5 2 2.5 3 T / T C How narrow is the sQGP transition?

This Ansatz is good, but not good enough! The interaction measure not rising steep enough: the maximum is displaced to much too high T We need another parameter to fix this: V nonpert → V nonpert − c ( M / T ) 2 ( | q | 2 ( 1 − | q | ) 2 1.5 Ext. model A 1 0.5 0 p/T 4 e/3T 4 ∆ /T 4 Latt. p/T 4 Latt. e/3T 4 Latt. ∆ /T 4 -0.5 1 1.5 2 2.5 3 T / T C How narrow is the sQGP transition?

1.5 1 0.5 0 p/T 4 e/3T 4 ∆ /T 4 Latt. p/T 4 Latt. e/3T 4 Latt. ∆ /T 4 -0.5 1 1.5 2 2.5 3 T / T C How narrow is the sQGP transition?

1.5 1 0.5 0 p/T 4 e/3T 4 ∆ /T 4 Latt. p/T 4 Latt. e/3T 4 Latt. ∆ /T 4 -0.5 1 1.5 2 2.5 3 T / T C SU(2) thermodynamic functions, c=2. How narrow is the sQGP transition?

SU(3),thermodynamic functions c=1 . How narrow is the sQGP transition?

Predictions The effective potential is now fixed. There are four predictions to be checked by lattice. Interface tension for T ≥ T c For SU(3) and higher N c : tension at T c for coexisting phases. Polyakov loop average How narrow is the sQGP transition?

Figure: Potential at T c , showing a a VERY weak first order transition, as function of 1 -q. q=1 is the confined state. You see a very small maximum at 1 − q = 0 . 16, i.e. 1 − q c = 0 . 33 is the minimum degenerate with the minimum at 1 − q = 0, the confining vacuum. How narrow is the sQGP transition?

Figure: Vertical blow up of the first graph and now you see the first order transition, i.e. the degeneracy at 1 − q = 0 and 1 − q = 0 . 33 How narrow is the sQGP transition?

Figure: Potential at 0.99 T c , as function of 1 -q. The metastable minimum at non-zero 1-q has almost gone away. How narrow is the sQGP transition?

For coexisting phases the tension is (SU(3)) = 0 . 0258012 T 2 g 2 ( T ) (smaller than Bursa -Teper � c / result). Large latent heat means very broad flat potential at T c 4 π 2 T 2 6 g 2 ( T ) ( 1 − ( T c / T ) 2 ) 3 / 2 if c=0, 3 √ SU(2) interface tension = and kinetic energy is taken classical. SU(2): the fit is done with (obviously) c=1.5 (SU(2) and just the two loop corrections from the complete QGP . The latter are not the whole story. We have to include loop corrections from fluctuations around the SQGP minima. This is being done. For SU(3) the tunneling path is only for the QGP along the λ 8 . Away from QGP there is a λ 3 component taken numerically into account, to obtain the minimal action. How narrow is the sQGP transition?

For coexisting phases the tension is (SU(3)) = 0 . 0258012 T 2 g 2 ( T ) (smaller than Bursa -Teper � c / result). Large latent heat means very broad flat potential at T c 4 π 2 T 2 6 g 2 ( T ) ( 1 − ( T c / T ) 2 ) 3 / 2 if c=0, 3 √ SU(2) interface tension = and kinetic energy is taken classical. SU(2): the fit is done with (obviously) c=1.5 (SU(2) and just the two loop corrections from the complete QGP . The latter are not the whole story. We have to include loop corrections from fluctuations around the SQGP minima. This is being done. For SU(3) the tunneling path is only for the QGP along the λ 8 . Away from QGP there is a λ 3 component taken numerically into account, to obtain the minimal action. How narrow is the sQGP transition?

For coexisting phases the tension is (SU(3)) = 0 . 0258012 T 2 g 2 ( T ) (smaller than Bursa -Teper � c / result). Large latent heat means very broad flat potential at T c 4 π 2 T 2 6 g 2 ( T ) ( 1 − ( T c / T ) 2 ) 3 / 2 if c=0, 3 √ SU(2) interface tension = and kinetic energy is taken classical. SU(2): the fit is done with (obviously) c=1.5 (SU(2) and just the two loop corrections from the complete QGP . The latter are not the whole story. We have to include loop corrections from fluctuations around the SQGP minima. This is being done. For SU(3) the tunneling path is only for the QGP along the λ 8 . Away from QGP there is a λ 3 component taken numerically into account, to obtain the minimal action. How narrow is the sQGP transition?

For coexisting phases the tension is (SU(3)) = 0 . 0258012 T 2 g 2 ( T ) (smaller than Bursa -Teper � c / result). Large latent heat means very broad flat potential at T c 4 π 2 T 2 6 g 2 ( T ) ( 1 − ( T c / T ) 2 ) 3 / 2 if c=0, 3 √ SU(2) interface tension = and kinetic energy is taken classical. SU(2): the fit is done with (obviously) c=1.5 (SU(2) and just the two loop corrections from the complete QGP . The latter are not the whole story. We have to include loop corrections from fluctuations around the SQGP minima. This is being done. For SU(3) the tunneling path is only for the QGP along the λ 8 . Away from QGP there is a λ 3 component taken numerically into account, to obtain the minimal action. How narrow is the sQGP transition?

For coexisting phases the tension is (SU(3)) = 0 . 0258012 T 2 g 2 ( T ) (smaller than Bursa -Teper � c / result). Large latent heat means very broad flat potential at T c 4 π 2 T 2 6 g 2 ( T ) ( 1 − ( T c / T ) 2 ) 3 / 2 if c=0, 3 √ SU(2) interface tension = and kinetic energy is taken classical. SU(2): the fit is done with (obviously) c=1.5 (SU(2) and just the two loop corrections from the complete QGP . The latter are not the whole story. We have to include loop corrections from fluctuations around the SQGP minima. This is being done. For SU(3) the tunneling path is only for the QGP along the λ 8 . Away from QGP there is a λ 3 component taken numerically into account, to obtain the minimal action. How narrow is the sQGP transition?

Interface SU(2), SU(3), de Forcrand, Noth, hep -lat/0506005 Interface tension ∼ (( T − T c ) / T c ) 3 / 2 , i.e. critical exponent =1.5 instead of universal 1.26. How narrow is the sQGP transition?

SU(3), Polyakov loop average, Gupta et al., arXiv:0711.2251 Narrowness of the sQGP ( T / T c = 1 to 1 . 2 ) ) is closely related to narrowness of interaction measure Our result does contradict the data. O ( g 2 ) corrections unlikely to produce agreement. Data without fuzzing the loop. How narrow is the sQGP transition?

SU(3), Polyakov loop average, Gupta et al., arXiv:0711.2251 Narrowness of the sQGP ( T / T c = 1 to 1 . 2 ) ) is closely related to narrowness of interaction measure Our result does contradict the data. O ( g 2 ) corrections unlikely to produce agreement. Data without fuzzing the loop. How narrow is the sQGP transition?

SU(3), Polyakov loop average, Gupta et al., arXiv:0711.2251 Narrowness of the sQGP ( T / T c = 1 to 1 . 2 ) ) is closely related to narrowness of interaction measure Our result does contradict the data. O ( g 2 ) corrections unlikely to produce agreement. Data without fuzzing the loop. How narrow is the sQGP transition?

0.4 SU(3) SU(4) 0.3 SU(6) Delta/(N^2-1) 0.2 0.1 0 1 1.5 2 2.5 3 3.5 4 T/Tc Figure: Interaction measure scaled by N 2 − 1, Panero 2009. Reduced discontinuity looks very small, like we find. How narrow is the sQGP transition?

0.14 0.12 0.10 0.08 0.06 4 5 6 7 8 9 10 Figure: Latent heat in units of ( N 2 − 1 ) T 4 c , Teper/Bursa: 0 . 744 − 0 . 34 / N 2 How narrow is the sQGP transition?

0.020 0.015 0.010 0.005 4 6 8 10 Figure: Order -disorder interface between coexisting phases, in units of ( N 2 − 1 ) T 2 g 2 ( T c ) N , as function of N � c / How narrow is the sQGP transition?

0.60 0.55 0.50 4 5 6 7 8 9 10 Figure: Jump of the normalized Polyakov loop at T c , as function of N How narrow is the sQGP transition?

Masses induced by the presence of the loop The presence of the loop induces a shift in the time derivatives, or equivalently in the Matsubara frequencies of off diagonal fluctuations: p 0 → p 0 + 2 π T ( q i − q j ) So the corresponding inverse propagator is corrected not only by m 2 ( q ) a q dependent Debye mass (O(gT)), but also p 2 + m 2 D ( q ) + ( 2 π T ( q i − q j )) 2 by an O(1) shift: � How narrow is the sQGP transition?

Illustration of the behaviour of the masses for SU(3). How narrow is the sQGP transition?

10 1 β =4.0760, T/T c =1.005 fit, large R 10 0 |F 1 (R)-F 1 ( ∞)|/Τ fit, interm. R 10 -1 10 -2 10 -3 10 -4 0 2 4 6 8 10 12 14 16 R/a = 4RT SU(3), How narrow is the sQGP transition?

Conclusions Model: EOS fully fixes effective potential Predicts surface tensions (o -o, o-d),Polyakov loop average, latent heat, Our model finds precocious QGP at T = 1 . 20 T c , beyond which P=1 The precociousness is persisting for more colours N c = 4 , 5 , .... If so: the Casimir scaling of the e-tension down to T ∼ 1 . 2 T c may be understandable, and should be compared to the Teper/Bursa lattice data. Conspicuously absent is prediction for magnetic tension Introduce quarks! How narrow is the sQGP transition?